1. Elementary Statistics
Thirteenth Edition
Chapter 2
Summarizing and Graphing Data
Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved

2. Summarizing and Graphing Data
2-1 Frequency Distributions for Organizing and Summarizing Data 2-2 Histograms 2-3 Graphs that Enlighten and Graphs that Deceive 2-4 Scatterplots, Correlation, and Regression

3. Key Concept
Introduce the analysis of paired sample data. Discuss correlation and the role of a graph called a scatterplot, and provide an introduction to the use of the linear correlation coefficient. Provide a very brief discussion of linear regression, which involves the equation and graph of the straight line that best fits the sample paired data.

4. Scatterplot and Correlation (1 of 2)
A correlation exists between two variables when the values of one variable are somehow associated with the values of the other variable.
Linear Correlation
A linear correlation exists between two variables when there is a correlation and the plotted points of paired data result in a pattern that can be approximated by a straight line.

5. Scatterplot and Correlation (2 of 2)
Scatterplot (or Scatter Diagram)
A scatterplot (or scatter diagram) is a plot of paired (x, y) quantitative data with a horizontal x-axis and a vertical y-axis. The horizontal axis is used for the first variable (x), and the vertical axis is used for the second variable (y).

6. Example: Waist and Arm Correlation (1 of 2)
Correlation: The distinct pattern of the plotted points suggests that there is a correlation between waist circumferences and arm circumferences.

7. Example: Waist and Arm Correlation (2 of 2)
No Correlation: The plotted points do not show a distinct pattern, so it appears that there is no correlation between weights and pulse rates.

8. Linear Correlation Coefficient r
The linear correlation coefficient is denoted by r, and it measures the strength of the linear association between two variables.

9. Using r for Determining Correlation
The computed value of the linear correlation coefficient, r, is always between −1 and 1.
If r is close to −1 or close to 1, there appears to be a correlation.
If r is close to 0, there does not appear to be a linear correlation.

10. Example: Correlation between Shoe Print Lengths and Heights? (1 of 2)
Shoe Print Length (cm)
29.7
31.4
31.8
27.6
Height (cm)
175.3
177.8
185.4
172.7

11. Example: Correlation between Shoe Print Lengths and Heights? (2 of 2)
It isn’t very clear whether there is a linear correlation.

12. P-Value
P-Value
If there really is no linear correlation between two variables, the P-value is the probability of getting paired sample data with a linear correlation coefficient r that is at least as extreme as the one obtained from the paired sample data.

13. Interpreting a P-Value from the Previous Example
The P-value of is high. It shows there is a high chance of getting a linear correlation coefficient of r = (or more extreme) by chance when there is no linear correlation between the two variables.

14. Interpreting a P-Value from the Example Where n = 5
Because the likelihood of getting r = or a more extreme value is so high (29.4% chance), we conclude there is not sufficient evidence to conclude there is a linear correlation between shoe print lengths and heights.

15. Interpreting a P-Value
Only a small P-value, such as 0.05 or less (or a 5% chance or less), suggests that the sample results are not likely to occur by chance when there is no linear correlation, so a small P-value supports a conclusion that there is a linear correlation between the two variables.

16. Example: Correlation between Shoe Print Lengths and Heights (n = 40)

17. Example: Correlation between Shoe Print Lengths and Heights
The scatterplot shows a distinct pattern. The value of the linear correlation coefficient is r = 0.813, and the P-value is Because the P-value of is small, we have sufficient evidence to conclude there is a linear correlation between shoe print lengths and heights.

18. Regression Regression
Given a collection of paired sample data, the regression line (or line of best fit, or least-squares line) is the straight line that “best” fits the scatterplot of the data.

19. Example: Regression Line (1 of 2)

20. Example: Regression Line (2 of 2)
The general form of the regression equation has a y-intercept of b0 = 80.9 and slope b1 = 3.22.
Using variable names, the equation is: Height = (Shoe Print Length)